On variational principle for upper metric mean dimension with potential

Abstract

Borrowing the idea of topological pressure determining measure-theoretical entropy in topological dynamical systems, we establish a variational principle for upper metric mean dimension with potential in terms of upper measure-theoretical metric mean dimension of invariant measures. Moreover, the notion of equilibrium state is introduced to characterize these measures that attain the supremum of the variational principle.

Final version. Many typos and mistakes are corrected. The main result was covered by the preprint [arXiv:2210.13126](Theorem 1.1), and was also coverd by Theorem A in [Carvalho, Pessil and Varandas, A convex analysis approach to the metric mean dimension: limits of scaled pressures and variational principles , Adv. Math. (436)(2024), Paper No. 109407, 54 pp.]